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Versions: 00 01 02 03 04 05 RFC 5510

Reliable Multicast Transport                                    J. Lacan
Internet-Draft                                          ENSICA/LAAS-CNRS
Intended status: Experimental                                    V. Roca
Expires: November 8, 2007                                          INRIA
                                                            J. Peltotalo
                                                            S. Peltotalo
                                        Tampere University of Technology
                                                             May 7, 2007


          Reed-Solomon Forward Error Correction (FEC) Schemes
                    draft-ietf-rmt-bb-fec-rs-03.txt

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   This Internet-Draft will expire on November 8, 2007.

Copyright Notice

   Copyright (C) The IETF Trust (2007).










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Abstract

   This document describes a Fully-Specified FEC Scheme for the Reed-
   Solomon forward error correction codes over GF(2^^m), with m in
   {2..16}, and its application to the reliable delivery of data objects
   on the packet erasure channel.

   This document also describes a Fully-Specified FEC Scheme for the
   special case of Reed-Solomon codes over GF(2^^8) when there is no
   encoding symbol group.

   Finally, in the context of the Under-Specified Small Block Systematic
   FEC Scheme (FEC Encoding ID 129), this document assigns an FEC
   Instance ID to the special case of Reed-Solomon codes over GF(2^^8).

   Reed-Solomon codes belong to the class of Maximum Distance Separable
   (MDS) codes, i.e. they enable a receiver to recover the k source
   symbols from any set of k received symbols.  The schemes described
   here are compatible with the implementation from Luigi Rizzo.
































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Table of Contents

   1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  4
   2.  Terminology  . . . . . . . . . . . . . . . . . . . . . . . . .  5
   3.  Definitions Notations and Abbreviations  . . . . . . . . . . .  6
     3.1.  Definitions  . . . . . . . . . . . . . . . . . . . . . . .  6
     3.2.  Notations  . . . . . . . . . . . . . . . . . . . . . . . .  6
     3.3.  Abbreviations  . . . . . . . . . . . . . . . . . . . . . .  7
   4.  Formats and Codes with FEC Encoding ID 2 . . . . . . . . . . .  8
     4.1.  FEC Payload ID . . . . . . . . . . . . . . . . . . . . . .  8
     4.2.  FEC Object Transmission Information  . . . . . . . . . . .  9
       4.2.1.  Mandatory Elements . . . . . . . . . . . . . . . . . .  9
       4.2.2.  Common Elements  . . . . . . . . . . . . . . . . . . .  9
       4.2.3.  Scheme-Specific Elements . . . . . . . . . . . . . . .  9
       4.2.4.  Encoding Format  . . . . . . . . . . . . . . . . . . . 10
   5.  Formats and Codes with FEC Encoding ID 5 . . . . . . . . . . . 12
     5.1.  FEC Payload ID . . . . . . . . . . . . . . . . . . . . . . 12
     5.2.  FEC Object Transmission Information  . . . . . . . . . . . 12
       5.2.1.  Mandatory Elements . . . . . . . . . . . . . . . . . . 12
       5.2.2.  Common Elements  . . . . . . . . . . . . . . . . . . . 12
       5.2.3.  Scheme-Specific Elements . . . . . . . . . . . . . . . 13
       5.2.4.  Encoding Format  . . . . . . . . . . . . . . . . . . . 13
   6.  Procedures with FEC Encoding IDs 2 and 5 . . . . . . . . . . . 14
     6.1.  Determining the Maximum Source Block Length (B)  . . . . . 14
     6.2.  Determining the Number of Encoding Symbols of a Block  . . 14
   7.  Small Block Systematic FEC Scheme (FEC Encoding ID 129)
       and Reed-Solomon Codes over GF(2^^8) . . . . . . . . . . . . . 16
   8.  Reed-Solomon Codes Specification for the Erasure Channel . . . 17
     8.1.  Finite Field . . . . . . . . . . . . . . . . . . . . . . . 17
     8.2.  Reed-Solomon Encoding Algorithm  . . . . . . . . . . . . . 18
       8.2.1.  Encoding Principles  . . . . . . . . . . . . . . . . . 18
       8.2.2.  Encoding Complexity  . . . . . . . . . . . . . . . . . 19
     8.3.  Reed-Solomon Decoding Algorithm  . . . . . . . . . . . . . 19
       8.3.1.  Decoding Principles  . . . . . . . . . . . . . . . . . 19
       8.3.2.  Decoding Complexity  . . . . . . . . . . . . . . . . . 20
     8.4.  Implementation for the Packet Erasure Channel  . . . . . . 20
   9.  Security Considerations  . . . . . . . . . . . . . . . . . . . 23
   10. IANA Considerations  . . . . . . . . . . . . . . . . . . . . . 24
   11. Acknowledgments  . . . . . . . . . . . . . . . . . . . . . . . 25
   12. References . . . . . . . . . . . . . . . . . . . . . . . . . . 26
     12.1. Normative References . . . . . . . . . . . . . . . . . . . 26
     12.2. Informative References . . . . . . . . . . . . . . . . . . 26
   Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 28
   Intellectual Property and Copyright Statements . . . . . . . . . . 29







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1.  Introduction

   The use of Forward Error Correction (FEC) codes is a classical
   solution to improve the reliability of multicast and broadcast
   transmissions.  The [2] document describes a general framework to use
   FEC in Content Delivery Protocols (CDP).  The companion document [4]
   describes some applications of FEC codes for content delivery.

   Recent FEC schemes like [9] and [10] proposed erasure codes based on
   sparse graphs/matrices.  These codes are efficient in terms of
   processing but not optimal in terms of correction capabilities when
   dealing with "small" objects.

   The FEC scheme described in this document belongs to the class of
   Maximum Distance Separable codes that are optimal in terms of erasure
   correction capability.  In others words, it enables a receiver to
   recover the k source symbols from any set of exactly k encoding
   symbols.  Even if the encoding/decoding complexity is larger than
   that of [9] or [10], this family of codes is very useful.

   Many applications dealing with content transmission or content
   storage already rely on packet-based Reed-Solomon codes.  In
   particular, many of them use the Reed-Solomon codec of Luigi Rizzo
   [7].  The goal of the present document to specify an implementation
   of Reed-Solomon codes that is compatible with this codec.

   The present document:

   o  introduces the Fully-Specified FEC Scheme with FEC Encoding ID 2
      that specifies the use of Reed-Solomon codes over GF(2^^m), with m
      in {2..16},

   o  introduces the Fully-Specified FEC Scheme with FEC Encoding ID 5
      that focuses on the special case of Reed-Solomon codes over
      GF(2^^8) and no encoding symbol group (i.e. exactly one symbol per
      packet), and

   o  in the context of the Under-Specified Small Block Systematic FEC
      Scheme (FEC Encoding ID 129) [3], assigns the FEC Instance ID 0 to
      the special case of Reed-Solomon codes over GF(2^^8) and no
      encoding symbol group.










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2.  Terminology

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
   document are to be interpreted as described in RFC 2119 [1].














































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3.  Definitions Notations and Abbreviations

3.1.  Definitions

   This document uses the same terms and definitions as those specified
   in [2].  Additionally, it uses the following definitions:

      Source symbol: unit of data used during the encoding process.

      Encoding symbol: unit of data generated by the encoding process.

      Repair symbol: encoding symbol that is not a source symbol.

      Systematic code: FEC code in which the source symbols are part of
      the encoding symbols.

      Source block: a block of k source symbols that are considered
      together for the encoding.

      Encoding Symbol Group: a group of encoding symbols that are sent
      together within the same packet, and whose relationships to the
      source block can be derived from a single Encoding Symbol ID.

      Source Packet: a data packet containing only source symbols.

      Repair Packet: a data packet containing only repair symbols.

3.2.  Notations

   This document uses the following notations:

      L denotes the object transfer length in bytes.

      k denotes the number of source symbols in a source block.

      n_r denotes the number of repair symbols generated for a source
      block.

      n denotes the encoding block length, i.e. the number of encoding
      symbols generated for a source block.  Therefore: n = k + n_r.

      max_n denotes the maximum number of encoding symbols generated for
      any source block.

      B denotes the maximum source block length in symbols, i.e. the
      maximum number of source symbols per source block.





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      N denotes the number of source blocks into which the object shall
      be partitioned.

      E denotes the encoding symbol length in bytes.

      S denotes the symbol size in units of m-bit elements.  When m = 8,
      then S and E are equal.

      m defines the length of the elements in the finite field, in bits.
      In this document, m belongs to {2..16}.

      q defines the number of elements in the finite field.  We have: q
      = 2^^m in this specification.

      G denotes the number of encoding symbols per group, i.e. the
      number of symbols sent in the same packet.

      GM denotes the Generator Matrix of a Reed-Solomon code.

      rate denotes the "code rate", i.e. the k/n ratio.

      a^^b denotes a raised to the power b.

      a^^-1 denotes the inverse of a.

      I_k denotes the k*k identity matrix.

3.3.  Abbreviations

   This document uses the following abbreviations:

      ESI stands for Encoding Symbol ID.

      FEC OTI stands for FEC Object Transmission Information.

      RS stands for Reed-Solomon.

      MDS stands for Maximum Distance Separable code.

      GF(q) denotes a finite field (A.K.A. Galois Field) with q
      elements.  We assume that q = 2^^m in this document.










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4.  Formats and Codes with FEC Encoding ID 2

   This section introduces the formats and codes associated to the
   Fully-Specified FEC Scheme with FEC Encoding ID 2 that specifies the
   use of Reed-Solomon codes over GF(2^^m).

4.1.  FEC Payload ID

   The FEC Payload ID is composed of the Source Block Number and the
   Encoding Symbol ID.  The length of these two fields depends on the
   parameter m (which is transmitted in the FEC OTI) as follows:

   o  The Source Block Number (field of size 32-m bits) identifies from
      which source block of the object the encoding symbol(s) in the
      payload is(are) generated.  There are a maximum of 2^^(32-m)
      blocks per object.

   o  The Encoding Symbol ID (field of size m bits) identifies which
      specific encoding symbol(s) generated from the source block
      is(are) carried in the packet payload.  There are a maximum of
      2^^m encoding symbols per block.  The first k values (0 to k - 1)
      identify source symbols, the remaining n-k values identify repair
      symbols.

   There MUST be exactly one FEC Payload ID per source or repair packet.
   In case of an Encoding Symbol Group, when multiple encoding symbols
   are sent in the same packet, the FEC Payload ID refers to the first
   symbol of the packet.  The other symbols can be deduced from the ESI
   of the first symbol by incrementing sequentially the ESI.

   The format of the FEC Payload ID for m = 8 and m = 16 is illustrated
   in Figure 1 and Figure 2 respectively.

    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |     Source Block Number (32-8=24 bits)        | Enc. Symb. ID |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

       Figure 1: FEC Payload ID encoding format for m = 8 (default)


    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | Src Block Nb (32-16=16 bits)  |  Enc. Symbol ID (m=16 bits)   |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+




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            Figure 2: FEC Payload ID encoding format for m = 16

4.2.  FEC Object Transmission Information

4.2.1.  Mandatory Elements

   o  FEC Encoding ID: the Fully-Specified FEC Scheme described in this
      section uses FEC Encoding ID 2.

4.2.2.  Common Elements

   The following elements MUST be defined with the present FEC scheme:

   o  Transfer-Length (L): a non-negative integer indicating the length
      of the object in bytes.  There are some restrictions on the
      maximum Transfer-Length that can be supported:

         max_transfer_length = 2^^(32-m) * B * E

      For instance, for m = 8, for B = 2^^8 - 1 (because the codec
      operates on a finite field with 2^^8 elements) and if E = 1024
      bytes, then the maximum transfer length is approximately equal to
      2^^42 bytes (i.e. 4 Tera Bytes).  Similarly, for m = 16, for B =
      2^^16 - 1 and if E = 1024 bytes, then the maximum transfer length
      is also approximately equal to 2^^42 bytes.  For larger objects,
      another FEC scheme, with a larger Source Block Number field in the
      FEC Payload ID, could be defined.  Another solution consists in
      fragmenting large objects into smaller objects, each of them
      complying with the above limits.

   o  Encoding-Symbol-Length (E): a non-negative integer indicating the
      length of each encoding symbol in bytes.

   o  Maximum-Source-Block-Length (B): a non-negative integer indicating
      the maximum number of source symbols in a source block.

   o  Max-Number-of-Encoding-Symbols (max_n): a non-negative integer
      indicating the maximum number of encoding symbols generated for
      any source block.

   Section 6 explains how to derive the values of each of these
   elements.

4.2.3.  Scheme-Specific Elements

   The following element MUST be defined with the present FEC Scheme.
   It contains two distinct pieces of information:




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   o  G: a non-negative integer indicating the number of encoding
      symbols per group used for the object.  The default value is 1,
      meaning that each packet contains exactly one symbol.  When no G
      parameter is communicated to the decoder, then this latter MUST
      assume that G = 1.

   o  Finite Field parameter, m: The m parameter is the length of the
      finite field elements, in bits.  It also characterizes the number
      of elements in the finite field: q = 2^^m elements.  The default
      value is m = 8.  When no finite field size parameter is
      communicated to the decoder, then this latter MUST assume that m =
      8.

4.2.4.  Encoding Format

   This section shows the two possible encoding formats of the above FEC
   OTI.  The present document does not specify when one encoding format
   or the other should be used.

4.2.4.1.  Using the General EXT_FTI Format

   The FEC OTI binary format is the following, when the EXT_FTI
   mechanism is used (e.g. within the ALC [11] or NORM [12] protocols).

    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |   HET = 64    |    HEL = 4    |                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               +
   |                      Transfer-Length (L)                      |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |       m       |       G       |   Encoding Symbol Length (E)  |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |  Max Source Block Length (B)  |  Max Nb Enc. Symbols (max_n)  |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

                      Figure 3: EXT_FTI Header Format

4.2.4.2.  Using the FDT Instance (FLUTE specific)

   When it is desired that the FEC OTI be carried in the FDT Instance of
   a FLUTE session [13], the following XML attributes must be described
   for the associated object:

   o  FEC-OTI-FEC-Encoding-ID

   o  FEC-OTI-Transfer-Length (L)




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   o  FEC-OTI-Encoding-Symbol-Length (E)

   o  FEC-OTI-Maximum-Source-Block-Length (B)

   o  FEC-OTI-Max-Number-of-Encoding-Symbols (max_n)

   o  FEC-OTI-Scheme-Specific-Info

   The FEC-OTI-Scheme-Specific-Info contains the string resulting from
   the Base64 encoding (in the XML Schema xs:base64Binary sense) of the
   following value:

    0                   1
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |       m       |       G       |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

    Figure 4: FEC OTI Scheme Specific Information to be included in the
                               FDT Instance

   When no m parameter is to be carried in the FEC OTI, the m field is
   set to 0 (which is not a valid seed value).  Otherwise the m field
   contains a valid value as explained in Section 4.2.3.  Similarly,
   when no G parameter is to be carried in the FEC OTI, the G field is
   set to 0 (which is not a valid seed value).  Otherwise the G field
   contains a valid value as explained in Section 4.2.3.  When neither m
   nor G are to be carried in the FEC OTI, then the sender simply omits
   the FEC-OTI-Scheme-Specific-Info attribute.

   After Base64 encoding, the 2 bytes of the FEC OTI Scheme Specific
   Information are transformed into a string of 4 printable characters
   (in the 64-character alphabet) and added to the FEC-OTI-Scheme-
   Specific-Info attribute.

















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5.  Formats and Codes with FEC Encoding ID 5

   This section introduces the formats and codes associated to the
   Fully-Specified FEC Scheme with FEC Encoding ID 5 that focuses on the
   special case of Reed-Solomon codes over GF(2^^8) and no encoding
   symbol group.

5.1.  FEC Payload ID

   The FEC Payload ID is composed of the Source Block Number and the
   Encoding Symbol ID:

   o  The Source Block Number (24 bit field) identifies from which
      source block of the object the encoding symbol in the payload is
      generated.  There are a maximum of 2^^24 blocks per object.

   o  The Encoding Symbol ID (8 bit field) identifies which specific
      encoding symbol generated from the source block is carried in the
      packet payload.  There are a maximum of 2^^8 encoding symbols per
      block.  The first k values (0 to k - 1) identify source symbols,
      the remaining n-k values identify repair symbols.

   There MUST be exactly one FEC Payload ID per source or repair packet.
   This FEC Payload ID refer to the one and only symbol of the packet.

    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |        Source Block Number (24 bits)          | Enc. Symb. ID |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

      Figure 5: FEC Payload ID encoding format with FEC Encoding ID 5

5.2.  FEC Object Transmission Information

5.2.1.  Mandatory Elements

   o  FEC Encoding ID: the Fully-Specified FEC Scheme described in this
      section uses FEC Encoding ID 5.

5.2.2.  Common Elements

   The Common Elements are the same as those specified in Section 4.2.2
   when m = 8 and G = 1.







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5.2.3.  Scheme-Specific Elements

   No Scheme-Specific elements are defined by this FEC Scheme.

5.2.4.  Encoding Format

   This section shows the two possible encoding formats of the above FEC
   OTI.  The present document does not specify when one encoding format
   or the other should be used.

5.2.4.1.  Using the General EXT_FTI Format

   The FEC OTI binary format is the following, when the EXT_FTI
   mechanism is used (e.g. within the ALC [11] or NORM [12] protocols).

    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |   HET = 64    |    HEL = 3    |                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               +
   |                      Transfer-Length (L)                      |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |   Encoding Symbol Length (E)  | MaxBlkLen (B) |     max_n     |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

          Figure 6: EXT_FTI Header Format with FEC Encoding ID 5

5.2.4.2.  Using the FDT Instance (FLUTE specific)

   When it is desired that the FEC OTI be carried in the FDT Instance of
   a FLUTE session [13], the following XML attributes must be described
   for the associated object:

   o  FEC-OTI-FEC-Encoding-ID

   o  FEC-OTI-Transfer-Length (L)

   o  FEC-OTI-Encoding-Symbol-Length (E)

   o  FEC-OTI-Maximum-Source-Block-Length (B)

   o  FEC-OTI-Max-Number-of-Encoding-Symbols (max_n)









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6.  Procedures with FEC Encoding IDs 2 and 5

   This section defines procedures that are common to FEC Encoding IDs 2
   and 5.  In case of FEC Encoding ID 5, m = 8 and G = 1.  Note that the
   block partitioning algorithm is defined in [2].

6.1.  Determining the Maximum Source Block Length (B)

   The finite field size parameter, m, defines the number of non zero
   elements in this field which is equal to: q - 1 = 2^^m - 1.  Note
   that q - 1 is also the theoretical maximum number of encoding symbols
   that can be produced for a source block.  For instance, when m = 8
   (default):

      max1_B = 2^^8 - 1 = 255

   Additionally, a codec MAY impose other limitations on the maximum
   block size.  Yet it is not expected that such limits exist when using
   the default m = 8 value.  This decision MUST be clarified at
   implementation time, when the target use case is known.  This results
   in a max2_B limitation.

   Then, B is given by:

      B = min(max1_B, max2_B)

   Note that this calculation is only required at the coder, since the B
   parameter is communicated to the decoder through the FEC OTI.

6.2.  Determining the Number of Encoding Symbols of a Block

   The following algorithm, also called "n-algorithm", explains how to
   determine the actual number of encoding symbols for a given block.

   AT A SENDER:

   Input:

      B: Maximum source block length, for any source block.  Section 6.1
      explains how to determine its value.

      k: Current source block length.  This parameter is given by the
      block partitioning algorithm.

      rate: FEC code rate, which is given by the user (e.g. when
      starting a FLUTE sending application).  It is expressed as a
      floating point value.




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   Output:

      max_n: Maximum number of encoding symbols generated for any source
      block

      n: Number of encoding symbols generated for this source block

   Algorithm:

      max_n = floor(B / rate);

      if (max_n > 2^^m - 1) then return an error ("invalid code rate");

      n = floor(k * max_n / B);

   AT A RECEIVER:

   Input:

      B: Extracted from the received FEC OTI

      max_n: Extracted from the received FEC OTI

      k: Given by the block partitioning algorithm

   Output:

      n

   Algorithm:

      n = floor(k * max_n / B);

   Note that a Reed-Solomon decoder does not need to know the n value.
   Therefore the receiver part of the "n-algorithm" is not necessary
   from the Reed-Solomon decoder point of view.  Yet a receiving
   application using the Reed-Solomon FEC scheme will sometimes need to
   know the n value used by the sender, for instance for memory
   management optimizations.  To that purpose, the FEC OTI carries all
   the parameters needed for a receiver to execute the above algorithm.











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7.  Small Block Systematic FEC Scheme (FEC Encoding ID 129) and Reed-
    Solomon Codes over GF(2^^8)

   In the context of the Under-Specified Small Block Systematic FEC
   Scheme (FEC Encoding ID 129) [3], this document assigns the FEC
   Instance ID 0 to the special case of Reed-Solomon codes over GF(2^^8)
   and no encoding symbol group.

   The FEC Instance ID 0 uses the Formats and Codes specified in [3].

   The FEC Scheme with FEC Instance ID 0 MAY use the algorithm defined
   in Section 9.1. of [3] to partition the file into source blocks.
   This FEC Scheme MAY also use another algorithm.  For instance the CDP
   sender may change the length of each source block dynamically,
   depending on some external criteria (e.g. to adjust the FEC coding
   rate to the current loss rate experienced by NORM receivers) and
   inform the CDP receivers of the current block length by means of the
   EXT_FTI mechanism.  This choice is out of the scope of the current
   document.
































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8.  Reed-Solomon Codes Specification for the Erasure Channel

   Reed-Solomon (RS) codes are linear block codes.  They also belong to
   the class of MDS codes.  A [n,k]-RS code encodes a sequence of k
   source elements defined over a finite field GF(q) into a sequence of
   n encoding elements, where n is upper bounded by q - 1.  The
   implementation described in this document is based on a generator
   matrix built from a Vandermonde matrix put into systematic form.

   Section 8.1 to Section 8.3 specify the [n,k]-RS codes when applied to
   m-bit elements, and Section 8.4 the use of [n,k]-RS codes when
   applied to symbols composed of several m-bit elements, which is the
   target of this specification.

8.1.  Finite Field

   A finite field GF(q) is defined as a finite set of q elements which
   has a structure of field.  It contains necessarily q = p^^m elements,
   where p is a prime number.  With packet erasure channels, p is always
   set to 2.  The elements of the field GF(2^^m) can be represented by
   polynomials with binary coefficients (i.e. over GF(2)) of degree
   lower or equal than m-1.  The polynomials can be associated to binary
   vectors of length m.  For example, the vector (11001) represents the
   polynomial 1 + x + x^^4.  This representation is often called
   polynomial representation.  The addition between two elements is
   defined as the addition of binary polynomials in GF(2) and the
   multiplication is the multiplication modulo a given irreducible
   polynomial over GF(2) of degree m with coefficients in GF(2).  Note
   that all the roots of this polynomial are in GF(2^^m) but not in
   GF(2).

   A finite field GF(2^^m) is completely characterized by the
   irreducible polynomial.  The following polynomials are chosen to
   represent the field GF(2^^m), for m varying from 2 to 16:

      m = 2, "111" (1+x+x^^2)

      m = 3, "1101", (1+x+x^^3)

      m = 4, "11001", (1+x+x^^4)

      m = 5, "101001", (1+x^^2+x^^5)

      m = 6, "1100001", (1+x+x^^6)

      m = 7, "10010001", (1+x^^3+x^^7)





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      m = 8, "101110001", (1+x^^2+x^^3+x^^4+x^^8)

      m = 9, "1000100001", (1+x^^4+x^^9)

      m = 10, "10010000001", (1+x^^3+x^^10)

      m = 11, "101000000001", (1+x^^2+x^^11)

      m = 12, "1100101000001", (1+x+x^^4+x^^6+x^^12)

      m = 13, "11011000000001", (1+x+x^^3+x^^4+x^^13)

      m = 14, "110000100010001", (1+x+x^^6+x^^10+x^^14)

      m = 15, "1100000000000001", (1+x+x^^15)

      m = 16, "11010000000010001", (1+x+x^^3+x^^12+x^^16)

   In order to facilitate the implementation, these polynomials are also
   primitive.  This means that any element of GF(2^^m) can be expressed
   as a power of a given root of this polynomial.  These polynomials are
   also chosen so that they contain the minimum number of monomials.

8.2.  Reed-Solomon Encoding Algorithm

8.2.1.  Encoding Principles

   Let s = (s_0, ..., s_{k-1}) be a source vector of k elements over
   GF(2^^m).  Let e = (e_0, ..., e_{n-1}) be the corresponding encoding
   vector of n elements over GF(2^^m).  Being a linear code, encoding is
   performed by multiplying the source vector by a generator matrix, GM,
   of k rows and n columns over GF(2^^m).  Thus:

      e = s * GM.

   The definition of the generator matrix completely characterizes the
   RS code.

   Let us consider that: n = 2^^m - 1 and: 0 < k <= n.  Let us denote by
   alpha the root of the primitive polynomial of degree m chosen in the
   list of Section 8.1 for the corresponding value of m.  Let us
   consider a Vandermonde matrix of k rows and n columns, denoted by
   V_{k,n}, and built as follows: the {i, j} entry of V_{k,n} is v_{i,j}
   = alpha^^(i*j), where 0 <= i <= k - 1 and 0 <= j <= n - 1.  This
   matrix generates a MDS code.  However, this MDS code is not
   systematic, which is a problem for many networking applications.  To
   obtain a systematic matrix (and code), the simplest solution consists
   in considering the matrix V_{k,k} formed by the first k columns of



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   V_{k,n}, then to invert it and to multiply this inverse by V_{k,n}.
   Clearly, the product V_{k,k}^^-1 * V_{k,n} contains the identity
   matrix I_k on its first k columns, meaning that the first k encoding
   elements are equal to source elements.  Besides the associated code
   keeps the MDS property.

   Therefore, the generator matrix of the code considered in this
   document is:

      GM = (V_{k,k}^^-1) * V_{k,n}

   Note that, in practice, the [n,k]-RS code can be shortened to a
   [n',k]-RS code, where k <= n' < n, by considering the sub-matrix
   formed by the n' first columns of GM.

8.2.2.  Encoding Complexity

   Encoding can be performed by first pre-computing GM and by
   multiplying the source vector (k elements) by GM (k rows and n
   columns).  The complexity of the pre-computation of the generator
   matrix can be estimated as the complexity of the multiplication of
   the inverse of a Vandermonde matrix by n-k vectors (i.e. the last n-k
   columns of V_{k,n}).  Since the complexity of the inverse of a k*k-
   Vandermonde matrix by a vector is O(k * log^^2(k)), the generator
   matrix can be computed in 0((n-k)* k * log^^2(k)) operations.  When
   the generator matrix is pre-computed, the encoding needs k operations
   per repair element (vector-matrix multiplication).

   Encoding can also be performed by first computing the product s *
   V_{k,k}^^-1 and then by multiplying the result with V_{k,n}.  The
   multiplication by the inverse of a square Vandermonde matrix is known
   as the interpolation problem and its complexity is O(k * log^^2 (k)).
   The multiplication by a Vandermonde matrix, known as the multipoint
   evaluation problem, requires O((n-k) * log(k)) by using Fast Fourier
   Transform, as explained in [14].  The total complexity of this
   encoding algorithm is then O((k/(n-k)) * log^^2(k) + log(k))
   operations per repair element.

8.3.  Reed-Solomon Decoding Algorithm

8.3.1.  Decoding Principles

   The Reed-Solomon decoding algorithm for the erasure channel allows
   the recovery of the k source elements from any set of k received
   elements.  It is based on the fundamental property of the generator
   matrix which is such that any k*k-submatrix is invertible (see [8]).
   The first step of the decoding consists in extracting the k*k
   submatrix of the generator matrix obtained by considering the columns



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   corresponding to the received elements.  Indeed, since any encoding
   element is obtained by multiplying the source vector by one column of
   the generator matrix, the received vector of k encoding elements can
   be considered as the result of the multiplication of the source
   vector by a k*k submatrix of the generator matrix.  Since this
   submatrix is invertible, the second step of the algorithm is to
   invert this matrix and to multiply the received vector by the
   obtained matrix to recover the source vector.

8.3.2.  Decoding Complexity

   The decoding algorithm described previously includes the matrix
   inversion and the vector-matrix multiplication.  With the classical
   Gauss-Jordan algorithm, the matrix inversion requires O(k^^3)
   operations and the vector-matrix multiplication is performed in
   O(k^^2) operations.

   This complexity can be improved by considering that the received
   submatrix of GM is the product between the inverse of a Vandermonde
   matrix (V_(k,k)^^-1) and another Vandermonde matrix (denoted by V'
   which is a submatrix of V_(k,n)).  The decoding can be done by
   multiplying the received vector by V'^^-1 (interpolation problem with
   complexity O( k * log^^2(k)) ) then by V_{k,k} (multipoint evaluation
   with complexity O(k * log(k))).  The global decoding complexity is
   then O(log^^2(k)) operations per source element.

8.4.  Implementation for the Packet Erasure Channel

   In a packet erasure channel, each packet (and symbol(s) since packets
   contain G >= 1 symbols) is either correctly received or erased.  The
   location of the erased symbols in the sequence of symbols MUST be
   known.  The following specification describes the use of Reed-Solomon
   codes for generating redundant symbols from the k source symbols and
   for recovering the source symbols from any set of k received symbols.

   The k source symbols of a source block are assumed to be composed of
   S m-bit elements.  Each m-bit element corresponds to an element of
   the finite field GF(2^^m) through the polynomial representation
   (Section 8.1).  If some of the source symbols contain less than S
   elements, they MUST be virtually padded with zero elements (it can be
   the case for the last symbol of the last block of the object).
   However, this padding need not be actually sent with the data to the
   receivers.

   The encoding process produces n encoding symbols of size S m-bit
   elements, of which k are source symbols (this is a systematic code)
   and n-k are repair symbols (Figure 7).  The m-bit elements of the
   repair symbols are calculated using the corresponding m-bit elements



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   of the source symbol set.  A logical j-th source vector, comprised of
   the j-th elements from the set of source symbols, is used to
   calculate a j-th encoding vector.  This j-th encoding vector then
   provides the j-th elements for the set encoding symbols calculated
   for the block.  As a systematic code, the first k encoding symbols
   are the same as the k source symbols and the last n-k repair symbols
   are the result of the Reed Solomon encoding.

        Input:  k source symbols

  0             j                                  S-1
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |             |X|                                   | source symbol 0
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |             |X|                                   | source symbol 1
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
              . . .
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |             |X|                                   | source symbol k-1
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

                *

        +----------------+
        |  generator     |
        |    matrix      |
        |      GM        |
        |   (k x n)      |
        +----------------+

                |
                V

      Output: n encoding symbols (source + repair)

  0             j                                  S-1
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |             |X|                                   | enc. symbol 0
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |             |X|                                   | enc. symbol 1
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
              . . .
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |             |Y|                                   | enc. symbol n-1
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+




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                     Figure 7: Packet encoding scheme

   An asset of this scheme is that the loss of some encoding symbols
   produces the same erasure pattern for each of the S encoding vectors.
   It follows that the matrix inversion must be done only once and will
   be used by all the S encoding vectors.  For large S, this matrix
   inversion cost becomes negligible in front of the S matrix-vector
   multiplications.

   Another asset is that the n-k repair symbols can be produced on
   demand.  For instance, a sender can start by producing a limited
   number of repair symbols and later on, depending on the observed
   erasures on the channel, decide to produce additional repair symbols,
   up to the n-k upper limit.  Indeed, to produce the repair symbol e_j,
   where k <= j < n, it is sufficient to multiply the S source vectors
   with column j of GM.



































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9.  Security Considerations

   Data delivery can be subject to denial-of-service attacks by
   attackers which send corrupted packets that are accepted as
   legitimate by receivers.  This is particularly a concern for
   multicast delivery because a corrupted packet may be injected into
   the session close to the root of the multicast tree, in which case
   the corrupted packet will arrive at many receivers.  This is
   particularly a concern for the FEC building block because the use of
   even one corrupted packet containing encoding data may result in the
   decoding of an object that is completely corrupted and unusable.  It
   is thus RECOMMENDED that source authentication and integrity checking
   are applied to decoded objects before delivering objects to an
   application.  For example, a SHA-1 hash [5] of an object may be
   appended before transmission, and the SHA-1 hash is computed and
   checked after the object is decoded but before it is delivered to an
   application.  Source authentication SHOULD be provided, for example
   by including a digital signature verifiable by the receiver computed
   on top of the hash value.  It is also RECOMMENDED that a packet
   authentication protocol such as TESLA [6] be used to detect and
   discard corrupted packets upon arrival.  Furthermore, it is
   RECOMMENDED that Reverse Path Forwarding checks be enabled in all
   network routers and switches along the path from the sender to
   receivers to limit the possibility of a bad agent successfully
   injecting a corrupted packet into the multicast tree data path.

   Another security concern is that some FEC information may be obtained
   by receivers out-of-band in a session description, and if the session
   description is forged or corrupted then the receivers will not use
   the correct protocol for decoding content from received packets.  To
   avoid these problems, it is RECOMMENDED that measures be taken to
   prevent receivers from accepting incorrect session descriptions,
   e.g., by using source authentication to ensure that receivers only
   accept legitimate session descriptions from authorized senders.

















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10.  IANA Considerations

   Values of FEC Encoding IDs and FEC Instance IDs are subject to IANA
   registration.  For general guidelines on IANA considerations as they
   apply to this document, see [2].

   This document assigns the Fully-Specified FEC Encoding ID 2 under the
   "ietf:rmt:fec:encoding" name-space to "Reed-Solomon Codes over
   GF(2^^m)".

   This document assigns the Fully-Specified FEC Encoding ID 5 under the
   "ietf:rmt:fec:encoding" name-space to "Reed-Solomon Codes over
   GF(2^^8)".

   This document assigns the FEC Instance ID 0 scoped by the Under-
   Specified FEC Encoding ID 129 to "Reed-Solomon Codes over GF(2^^8)".
   More specifically, under the "ietf:rmt:fec:encoding:instance" sub-
   name-space that is scoped by the "ietf:rmt:fec:encoding" called
   "Small Block Systematic FEC Codes", this document assigns FEC
   Instance ID 0 to "Reed-Solomon Codes over GF(2^^8)".































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11.  Acknowledgments

   The authors want to thank Brian Adamson for his valuable comments.
   The authors also want to thank Luigi Rizzo for comments on the
   subject and for the design of the reference Reed-Solomon codec.














































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12.  References

12.1.  Normative References

   [1]   Bradner, S., "Key words for use in RFCs to Indicate Requirement
         Levels", RFC 2119.

   [2]   Watson, M., Luby, M., and L. Vicisano, "Forward Error
         Correction (FEC) Building Block",
          draft-ietf-rmt-fec-bb-revised-07.txt (work in progress),
         April 2007.

   [3]   Watson, M., "Basic Forward Error Correction (FEC) Schemes",
          draft-ietf-rmt-bb-fec-basic-schemes-revised-03.txt (work in
         progress), February 2007.

   [4]   Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley, M.,
         and J. Crowcroft, "The Use of Forward Error Correction (FEC) in
         Reliable Multicast", RFC 3453, December 2002.

   [5]   "HMAC: Keyed-Hashing for Message Authentication", RFC 2104,
         February 1997.

   [6]   "Timed Efficient Stream Loss-Tolerant Authentication (TESLA):
         Multicast Source Authentication Transform Introduction",
         RFC 4082, June 2005.

12.2.  Informative References

   [7]   Rizzo, L., "Reed-Solomon FEC codec (revised version of July
         2nd, 1998), available at
         http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz, and
         mirrored at http://planete-bcast.inrialpes.fr/", July 1998.

   [8]   Mac Williams, F. and N. Sloane, "The Theory of Error Correcting
         Codes", North Holland, 1977 .

   [9]   Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer,
         "Raptor Forward Error Correction Scheme",
          draft-ietf-rmt-bb-fec-raptor-object-08 (work in progress),
         April 2007.

   [10]  Roca, V., Neumann, C., and D. Furodet, "Low Density Parity
         Check (LDPC) Forward Error Correction",
          draft-ietf-rmt-bb-fec-ldpc-06.txt (work in progress),
         May 2007.

   [11]  Luby, M., Watson, M., and L. Vicisano, "Asynchronous Layered



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         Coding (ALC) Protocol Instantiation",
          draft-ietf-rmt-pi-alc-revised-04.txt (work in progress),
         February 2007.

   [12]  Adamson, B., Bormann, C., Handley, M., and J. Macker,
         "Negative-acknowledgment (NACK)-Oriented Reliable Multicast
         (NORM) Protocol",  draft-ietf-rmt-pi-norm-revised-04.txt (work
         in progress), March 2007.

   [13]  Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca,
         "FLUTE - File Delivery over Unidirectional Transport",
          draft-ietf-rmt-flute-revised-03.txt (work in progress),
         January 2007.

   [14]  Gohberg, I. and V. Olshevsky, "Fast algorithms with
         preprocessing for matrix-vector multiplication problems",
         Journal of Complexity, pp. 411-427, vol. 10, 1994 .


































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Authors' Addresses

   Jerome Lacan
   ENSICA/LAAS-CNRS
   1, place Emile Blouin
   Toulouse  31056
   France

   Email: jerome.lacan@ensica.fr
   URI:   http://dmi.ensica.fr/auteur.php3?id_auteur=5


   Vincent Roca
   INRIA
   655, av. de l'Europe
   Inovallee; Montbonnot
   ST ISMIER cedex  38334
   France

   Email: vincent.roca@inrialpes.fr
   URI:   http://planete.inrialpes.fr/~roca/


   Jani Peltotalo
   Tampere University of Technology
   P.O. Box 553 (Korkeakoulunkatu 1)
   Tampere  FIN-33101
   Finland

   Email: jani.peltotalo@tut.fi
   URI:   http://atm.tut.fi/mad


   Sami Peltotalo
   Tampere University of Technology
   P.O. Box 553 (Korkeakoulunkatu 1)
   Tampere  FIN-33101
   Finland

   Email: sami.peltotalo@tut.fi
   URI:   http://atm.tut.fi/mad










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Full Copyright Statement

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   This document is subject to the rights, licenses and restrictions
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